In the last few years, several extensions of OT comparing arbitrary positive measures have been developed, based on the different equivalent formulations of OT.
The first extensions in the literature focused on the Benamou-Brenier formulation [1]}, and relax a mass conservation constraint encoded by a continuity PDE [2]}, [3]}, [4]}, [5]}.
This review focuses mostly on the one proposed by [2]}, which consists in replacing the hard constraints \((p_)_\sharp \pi =\)  and \((p_)_\sharp \pi =\)  by \(\phi \) -divergences \({\phi _1}((p_)_\sharp \pi | )\)  and \({\phi _2}((p_)_\sharp \pi | )\)  (presented in Section REF ).
It was also proposed in [7]} for the particular setting \(\phi =\) .
We mainly focus on it because it is convenient from a computational perspective, as we detail in Section REF .
